Lie groups of real analytic diffeomorphisms are L1-regular

Abstract

Let $M$ be a compact, real analytic manifold and $G≔{Diff}^{\omega }\left(M\right)$ be the Lie group of all real-analytic diffeomorphisms $\gamma :M\to M$, which is modelled on the locally convex space $g≔{\Gamma }^{\omega }\left(TM\right)$ of real-analytic vector fields on $M$. Let $AC([0,1],G)$ be the Lie group of all absolutely continuous functions $\eta :[0,1]\to G$. We study flows of time-dependent real-analytic vector fields on $M$ which are $L^1$ in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group ${Diff}^{\omega }\left(M\right)$ is $L^1$-regular in the sense that each $\left[\gamma \right]\in L^1([0,1],g)$ has an evolution $Evol\left(\left[\gamma \right]\right)\in AC([0,1],G)$ which depends smoothly on $\left[\gamma \right]$. As tools for the proof, we develop new results concerning $L^1$-regularity of infinite-dimensional Lie groups, and new results concerning the continuity and complex analyticity of non-linear mappings on locally convex direct limits.